Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Probabiliy Distribution Proof Please help and provide suggestions.
(i) A discrete random variable X has the distribution $P(X = i) = 2a^i $ for i ∈ N+ (where N+ := {1,2,...}). What is the value of a?
$P(X = 1) = 2a^1 ;P(X = 2) = 2a^2 ;P(X = 1) = 2a^3 ;P(X = 1) = 2a^4;...... $
$S_n=\sum_{i=1}^\infty 2a^i=\frac{2a(1-a^n... | As $0\le 2a\le 1$ then $a^n\to 0$ as $n\to \infty$. So
$$1=S=\sum_{i=1}^\infty 2a^i=\frac{2a}{1-a}.$$
So $2a=1-a$ and then $a=\frac{1}{3}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving a recurrence relation (textbook question) $a_{n+1} - a_n = 3n^2 - n$ ;$a_0=3$
I need help for solving the particular solution.
Based on a chart in my textbook if you get $n^2$ the particular solution would be
$A_2n^2 + A_1n + A_0$ and $n$ has the particular solution of $A_1n+A_0$.
So given $3n^2 - n$, my first... | hint: $a_n = (a_n-a_{n-1})+(a_{n-1}-a_{n-2})+\cdots +(a_2-a_1)+(a_1-a_0)+a_0$
| {
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Is it true that every element of $V \otimes W$ is a simple tensor $v \otimes w$? I know that every vector in a tensor product $V \otimes W$ is a sum of simple tensors $v \otimes w$ with $v \in V$ and $w \in W$. In other words, any $u \in V \otimes W$ can be expressed in the form$$u = \sum_{i=1}^r v_i \otimes w_i$$for s... | This is equivalent to asking, does every multivariable polynomial factor into polynomials of one variable? No.
Consider the polynomial $x^2 + y$. If it could be factored into $P(x)Q(y)$ Then for some value of $y$, $Q(y) = 0$, and thus $x^2 + y = 0$ no matter the value of $x$. This is obviously false.
To be precise, le... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the order of an element in the dihedral group of order 4. How do I find the order of
$$S_1=\left({\begin{array}{cc}
\cos\frac{\pi}{3} & \sin\frac{\pi}{3}\\
\sin\frac{\pi}{3} & -\cos\frac{\pi}{3}\\
\end{array} }\right)$$
I know that $S_1$ is a dihedral group and is a reflection of the line that makes an... | $$S_1^2=\begin{bmatrix} 1&0\\0&1\end{bmatrix}$$
In more detail
$$\begin{align}S_1^2=&\begin{matrix} \cos^2{\frac{\pi}{3}}+\sin^2{\frac{\pi}{3}}&\cos{\frac{\pi}{3}}\sin{\frac{\pi}{3}}-\cos{\frac{\pi}{3}}\sin{\frac{\pi}{3}}\\\cos{\frac{\pi}{3}}\sin{\frac{\pi}{3}}-\cos{\frac{\pi}{3}}\sin{\frac{\pi}{3}}&\cos^2{\frac{\pi}{3... | {
"language": "en",
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Prove complements of independent events are independent. Given a finite set of events $\{A_i\}$ which are mutually independent, i.e., for every subset $\{A_n\}$,
$$\mathrm{P}\left(\bigcap_{i=1}^n A_i\right)=\prod_{i=1}^n \mathrm{P}(A_i).$$
show that the set $\{A_i^c\}$, that is the set of complements of the original e... | Hint: prove that the set of events stays independent if you replace one of them by its complement, i.e. that given your conditions the set $\{A_1^c, A_2, \ldots, A_n\}$ is independent. Then use this $n$ times to replace all of $A_i$ by their complements one by one.
Update. Hint 2: to avoid clutter, let me show you what... | {
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What is ⌊0.9 recurring ⌋? For a ceiling and floor function, the number is taken to 0 decimal places. Does this process mean that 0.9 recurring inside a floor function would go to 0? Or would the mathematician take 0.9 recurring to be equal to 1, thus making the answer 1?
And if 0.9 recurring does equal 1, does that mea... | You have to look at the $.9$ recurring as a sum... then you'll know the answer.
$$\bar{.9} = \sum_{i=1}^{\infty} \frac{9}{10^{i}}$$
So, $$\lfloor \bar{.9}\rfloor = \left\lfloor \sum_{i=1}^{\infty}\frac{9}{10^{i}}\right\rfloor=\lfloor 1 \rfloor = 1.$$ You cannot split up the floor function over a sum, i.e. $\lfloor a+b... | {
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Number Theory Taxicab Number How to prove that there are infinite taxicab numbers?
ok i was reading this http://en.wikipedia.org/wiki/Taxicab_number#Known_taxicab_numbers
and thought of this question..any ideas?
| It is easy to show that there are infinitely many positive integers which are representable as the sum of two cubes, e.g., see the article Characterizing the Sum of Two Cubes by K.A. Broughan (2003). If we require a representation as the sum of two cubes in at least $N\ge 2$ different ways, then the result is more diff... | {
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"question_score": "2",
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$(1-\zeta_m)$ is a unit in $\mathbb{Z}[\zeta_m]$ if m contains at least two prime factors We know that for $m=p^r, 1-\zeta_m$ is a prime.Now suppose that m has at least 2 distinct primes appearing in its prime factorization,we need to show that $1-\zeta_m$ is a unit in its ring of integers $\mathcal{O}_{\mathbb{Q}(\zet... | Write $1+x+x^2+...+x^{n-1} = \prod_{j=1}^{n-1} (x-\zeta_n^j)$ and put x=1 to get $n = \prod_{j=1}^{n-1} (1-\zeta_n^j)$. If $p^a||n$, then running $j$ through multiples of $n/p^a$, we see that the product contains $p^a = \prod_{j=1}^{p^a-1} (x-\zeta_{p^a}^j)$. Remove all such factors and get $1 = \prod (1 - \zeta_n^j)$ ... | {
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Is there a formal name for this matrix? I've been using a matrix of the following form:
$$
\begin{bmatrix}
-1 & 1 & 0 \\
0 & -1 & 1 \\
1 & 0 & -1
\end{bmatrix}
$$
Which is just a circularly-shifted $I$ matrix minus another $I$ matrix. Essentially, a permutation matrix minus an identity. Is there a formal name for that?... | Not that I have heard. It is a circulant matrix, though, and the permutation matrix alone without the $-I$ is sometimes called the cyclic shift matrix, circulant generator or generator of the circulant algebra.
| {
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Prove that $\sqrt[5]{672}$ is irrational How would you prove $\sqrt[5]{672}$ is irrational?
I was trying proof by contradiction starting by saying:
Suppose $\sqrt[5]{672}$ is rational ...
| $$672^{1/5}=\frac pq,$$($p$ and $q$ relative primes) then
$$p^5=672q^5,$$
which is possible only if $p$ is a multiple of $7$, which in turn implies that $q$ is a multiple of $7$.
| {
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Proving an identity involving differentials of integrals
Define $$ E(\theta,k) = \int_0^\theta \sqrt{1-k^2\sin{^2{x}}} dx$$ and $$F(\theta,k) = \int_0^\theta \frac{1}{\sqrt{1-k^2\sin{^2{x}}}} dx$$ We are to show $$\left(\frac{\partial E}{\partial k}\right)_\theta = \frac{E-F}{k} $$
Am I right in thinking: $$\left(\fr... | Starting from your correct assumption
$$\left(\frac{\partial E}{\partial k}\right)_\theta =\int_0^\theta \frac{\partial}{\partial k}\sqrt{1-k^2\sin{^2{x}}}\ dx$$
the differential, however, evaluates to
$$\begin{align}\left(\frac{\partial E}{\partial k}\right)_\theta &=\int_0^\theta \frac{-k\sin^2\theta}{\sqrt{1-k^2\sin... | {
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Calculate limit of $\sqrt[n]{2^n - n}$ Calculate limit of $\sqrt[n]{2^n - n}$.
I know that lim $\sqrt[n]{2^n - n} \le 2$, but don't know where to go from here.
| The exponential function is continuous, so $$\lim_{n\rightarrow \infty}\sqrt[n]{2^n -n} = \lim_{n\rightarrow \infty} e^{\frac{\ln(2^n-n)}{n}} = e^{(\lim_{n\rightarrow \infty}\frac{\ln(2^n-n)}{n})}$$
Then you could use l'Hospital to show that $$\lim_{n\rightarrow \infty}\frac{\ln(2^n-n)}{n} = \ln(2)$$
So then you'd have... | {
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Can we subtract a trigonometric term from a polynomial? Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations?
thanks.
| The answer to your posed problem cannot be expressed in general in terms of elementary formulas in closed form. So in practice, numerical root finding is the only way to go. Luckily, for your question finding a root approximately is not that hard using Newton's method.
| {
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I'm unsure which test to use for this series: $\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$? I want to determine if this series converges or diverges: $$\sum\limits_{n=1}^\infty{\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}}$$
I tried the Ratio Test at first, and didn't get anywhere with that. I'm thinking ... | You may write, as $n \to \infty$,
$$\frac{3^\frac{1}{n} \sqrt{n}}{2n^2-5}= \frac{\sqrt{n}}{2n^2}\frac{e^{\large \frac{\ln 3}{n}}}{1-\frac{5}{2n^2}}=\frac{1}{2}\frac{1}{n^{3/2}}\frac{1+\frac{\ln 3}{n}+\mathcal{O}\left(\frac {1}{n^2}\right)}{1-\frac{5}{2n^2}}\sim \frac{1}{2}\frac{1}{n^{3/2}}$$ and your initial series is ... | {
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Proving logical equivalences The question is to prove
$\neg (p \wedge q) \to (p \vee r)$ equivalent to $p \vee r$
So far, I got
*
*$¬[¬(p \wedge q)] \vee (p \vee r)$ - implication
*$(p \wedge q) \vee (p \vee r)$ - double negation
Now, is this question logically not equivalent?
Or is there some way I can... | $(p \land q) \lor (p \lor r)$ is logically equivalent to $(p \land q) \lor p \lor r$ (the parentheses can be removed because we have the same $\lor$ sign inside and outside the parentheses. This is in turn logically equivalent to $p \land q \lor r$ which implies $p \lor r$. But the converse is not true, so I don't thin... | {
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Verification - does $x < \sup A$ necessarily mean $x \in A$? Suppose we have a non-empty and bounded above set $A$, and some $x \in \mathbb{R}$ such that $x < \sup A$.
Do we then have that $x \in A$?
Since we assume $\sup A$ exists, it follows that $A \subset \mathbb{R}$.
If we take $x = \sqrt{2}$ and define $A = \{...... | There is no problem in your example. In fact you don't need to consider an infinite set as an example. You can just consider any finite set and $x < \min(A) \leq \max(A)$ but $x \notin A$.
| {
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Discrete subset of $\mathbb R^2$ such that $\mathbb R^2\setminus S$ is path connected. Let, $S\subset \mathbb R^2$ be defined by $$S=\left\{\left(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}\right):m,n,p,q\in \mathbb Z\right\}.$$ Then, which are correct?
(A) $S$ is a discrete set.
(B) $\mathbb R^2\setminus S$ is path connec... | Hint: Try to think of the graph of S near any of its limit points (m,n). It will somewhat look like kitchen sink filter which has more and more holes as you approach towards its centre (a limit point).
| {
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integration of definite integral involving sinx and cos x Evaluate $\int_0^{\pi}\frac{dx}{a^2\cos^2x +b^2 \sin^2x}$
I got numerator $\sec^2 x$ and denominator $b^2 ( a^2/b^2 + \tan^2x)$.
I made substitution $u= \tan x$. That way $\sec^2 x$ got cancelled and the answer was of form $1/ab$ ($\tan^{-1} (bu/a)$)
And then if... | Are you talking about this?:
$$\small\int\frac{dx}{a^2\cos^2x +b^2 \sin^2x}=\int\frac{\sec^2xdx}{a^2+b^2 \tan^2x}\stackrel{u=\tan x}=\frac1{b^2}\int\frac{du}{a^2/b^2+u^2}=\frac1{b^2}\frac1{a/b}\arctan\frac{\tan x}{a/b}$$
So:
$$\int_0^{\pi}\frac{dx}{a^2\cos^2x +b^2 \sin^2x}=\frac1{ab}\arctan\frac{b\tan x}a\Bigg|_0^... | {
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Proving $\sum_{i=0}^n 2^i=2^{n+1}-1$ by induction. Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for.
I must prove the following using mathematical induction:
For all $n\in\mathbb{Z^+}$, $1+2+2^2+2^3+\cdots+2^n=2^{n+1}-1$.
This is what ... | Your problem has already been answered in details. I would like to point out that if you want to train your inductive skills, once you have a solution to your problem, you still can explore it from many other sides, especially using visual proofs that can be quite effective. And find other proofs. Some are illustrated ... | {
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Is there a more general version of $row rank(A) = column rank(A)$? $A$ is a $m\times n$ matrix in field $F$ There's a conclusion in matrix that, given $A$ a $m\times n$ matrix in field $F$, one has $$row\, rank(A) = column\,rank(A)$$
Since linear algebra conclusions are sometimes related to more general ones in abstrac... | Depends on how general you want to get. If going from matrices to linear maps is general enough, then if you have $V$ and $W$ as finite-dimensional vector spaces over the same field $\mathbb{F}$, and if $A:V\rightarrow W$ is a linear map, we call the rank of $A$ the dimension of its range (which is as we know is a subs... | {
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Is it true that $2^n$ is $O(n!)$? I had a similar problem to this saying:
Is it true that $n!$ is $O(2^n)$?
I got that to be false because if we look at the dominant power of $n!$ it results in $n^n$. So because the base numbers are not the same it is false.
Is it true that $2^n$ is $O(n!)$?
So likewise with the bases,... | If $f(n)$ is $O(g(n))$, then there is a constant $C$ such that $f(n) \leq C g(n)$ eventually.
It turns out that $2^n$ is $O(n!)$. Can you find a constant $C$ and prove the inequality? Hint: Choose $C=2$ and try to prove $2^n \leq 2 n!$.
| {
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For any $n\ge2$ prove that $H(X_1,X_2,...,X_n)\ge\sum\limits_{i=n}^\mathbb{n}\ H(X_i|X_j , j \neq i)$ I am trying to figure this out and I am stuck. Any ideas?
For any $n\ge2$ prove that $H(X_1,X_2,\ldots,X_n)\ge\sum\limits_{i=1}^\mathbb{n}\ H(X_i\mid X_j , \ j \neq i)$
| For n=2 we have $$H(X1,X2)=H(X1)+H(X2|X1) \ge H(X_1|X_2)+ H(X_2|X_1) \ (1)$$ which stands because conditioning does not increase entropy.
Same logic for n>2
| {
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How is the gradient of a function $f$ equal to the Frechet derivative? A mapping $f$ from an open set $S \subset \mathbb{R^n}$ into $\mathbb{R^m}$ is said to be differentiable at $\vec{a} \in S$ if there is an $n\times m$ matrix $L$ such that $\lim_{\vec{h} \to 0}\dfrac{|f(\vec{a}+\vec{h})-f(\vec{a})-L\cdot \vec{h}|}{|... | Gradient of function usually means right away that you are restricting to $m=1$ (otherwise we'd be talking about a Jacobian matrix). But even then, it's preferable to call the linearized map $L$ you found as the differential, Frechet or total derivative before associating it to the gradient vector. Take a look at the f... | {
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Calculate the maximum area (maximum value) TX farmer has 100 metres of fencing to use to make a rectangular enclosure for sheep as shown.
He will use existing walls for two sides of the enclosure and leave an opening of 2 metres for a gate.
a) Show that the area of the enclosure is given by: $A = 102x – x^2.$
b) Fin... | From your picture, one side of the rectangle is x.
Since you have 100 metres, this means the other side has length (100 - x) + 2 = 102 - x.
So, the Area A = $(102 - x) * x = 102x - x^2$
The maximum area occurs where $\frac{dA}{dx} = 102 - 2x = 0$
or where $x = 51$
So, the max area A = $102(51) - 51^2$
| {
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Surfaces in $\mathbb P^3$ not containing any line Let $d \geq 4$. I'm interested by know if there is a surface $S$ of degree $d$ in $\mathbb P^3_{\mathbb C}$ such that $S$ does not contains a line. I know
I have no idea how to do it.
| This is the famous Noether-Lefschetz theorem. The answer is that for a "very general" such surface -- meaning away from a countable union of proper closed subsets in the parameter space of all degree-d surfaces -- the only algebraic curves are complete intersections with other surfaces. In particular, there are no line... | {
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Atiyah-Macdonald, Exercise 5.4 I was having some trouble with the following exercise from Atiyah-Macdonald.
Let $A$ be a subring of $B$ such that $B$ is integral over $A$. Let $\mathfrak{n}$ be a maximal ideal of $B$ and let $\mathfrak{m}=\mathfrak{n} \cap A$ be the corresponding maximal ideal of $A$. Is $B_{\mathfrak... | Maybe you already noticed that $\mathfrak n^c=(x^2-1)$. Now apply the definition of integrality and after clearing the denominators you get $\sum_{i=0}^n a_is_i(x+1)^{n-i}=0$ with $a_i\in A$, $a_n=1$, and $s_i\in A-\mathfrak n^c$. Then $x+1\mid s_n$ (in $B$), so $s_n\in (x+1)B\cap A=(x^2-1)$, a contradiction.
| {
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Prove there exists $m$ and $b$ such that $h(x) = mx + b$ Problem:
Suppose ∃ function h: ℝ → ℝ such that h as a second derivative h''(x) = 0 ∀ x ∈ ℝ. Prove ∃ numbers m, b:
h(x) = mx + b, ∀ x ∈ ℝ.
My attempt:
Consider h(x) = mx + b, with constants m and b. Note:
h(x) = mx + b
h'(x) = m + 0
h''(x) = 0.
Hence the state... | Because integration is the inverse of differentiation, we can find that if $h(x)=0,$
$$\int h ''(x) \,dx=\int 0\, dx\implies h'(x)=C_1\implies \int h'(x) \,dx=\int C_1 \,dx\implies h(x)=xC_1+C_2$$
Now we can choose $m=C_1$ and $b=C_2$, and we are done.
| {
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liminf inequality in measure spaces
Let $(X;\mathscr{M},\mu)$ be a measure space and $\{E_j\}_{j=1}^\infty\subset \mathscr{M}$. Show that $$\mu(\liminf E_j)\leq \liminf \mu(E_j)$$
and, if $\mu\left(\bigcup_{j=1}^\infty E_j\right)<\infty$, that
$$\mu(\limsup E_j)\geq \limsup \mu(E_j).$$
I'm trying to parse what's ... | $\left(\bigcap_{j=i}^\infty E_j\right)_{i=1}^\infty$ is an increasing sequence of sets, so you may have a theorem that states that $$\mu\left(\bigcup_{i=1}^\infty \bigcap_{j=i}^\infty E_j\right) = \lim_{i \to \infty} \mu\left(\bigcap_{j=i}^\infty E_j\right).$$ Then, note that $\mu\left(\bigcap_{j=i}^\infty E_j\right) \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1194833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Working with norms I was hoping to get some help with being able to properly work with norms and derivatives so I can actually understand my PDE course. We are currently working on Sobolev spaces.
Example, I want to show that:
$$-\int_{U} u \Delta u dx \leq C \int_{U}|u||D^2u|dx$$
$u \in C_{c}^{\infty}(U)$ with $U$ bo... | It looks like you're using a book using notation similar to Evans. If you are, check Appendix A, in the section Notation for derivatives, where we see
$$
|D^k u| = \left(\sum_{|\alpha|=k} |D^\alpha u|^2\right)^{1/2}.
$$
With this it should be pretty clear why $|\Delta u| \leq C|D^2 u|$.
Notation for derivatives is a bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1194935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculate Wronksian of Second Order Differential Equation Use variation of parameters to find a particular solution to:
$\frac{d^{2}y}{dx^{x}} + 2 \frac{dy}{dx} + y = \frac{1}{x^{4}e^{4}}.$
There are no solutions given so finding a wronskian that way is nil.
But since it is still in the order $p(x)y'' + q(x)y' + r(x)y ... | Since the discriminant of the differential equation $y'' + 2y' + y = 0$ is $2^{2} - 4 = 0,$ it follows that $$u_{1} := e^{-x},\ u_{2} := xe^{-x}$$ are the basis solutions. If $x \mapsto w$ is the Wronskain of $u_{1}$ and $u_{2}$, then
$$w = u_{1}u_{2}' - u_{2}u_{1}' = e^{-2x}.$$
Let $R(x) := 1/x^{4}e^{4},$ let $t_{1} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that if every nonempty open set in $X$ is non-meager, then every comeager set in $X$ is dense Suppose $X$ is a topological space. Prove that if every nonempty open set in $X$ is non-meager, then every comeager set in $X$ is dense.
My attempt:
Suppose $A \subset X$ is a comeager set and let $O \subset X$ be an open... | No. Although, for all $n$, the set $A_n\cap O$ is not empty, it might contain entirely different points for different $n$. So we may not conclude $\bigcap_n{A_n}\cap O$ nonempty. This is precisely why you need the unused hypothesis.
On the other hand, note that if $$\emptyset=\bigcap_n{A_n}\cap O$$ then, taking th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finite integral with removable singularity I wanted to integrate $\frac{(exp(-x) -1)^2}{x}$ from $x=0$ to $x=a$ where $a$ is finite. Since the integrand, viz., $\frac{(exp(-x) -1)^2}{x}$ has a removable singularity at $x=0$ , I can take the lower limit to be zero for the integration. Further, if I use finite integratio... | Probably the best way is to regularise the integral by adding an $x^s$, calculate it in terms of incomplete Gamma functions, then let $s \to 0$. We have
$$ \int_0^a x^{s-1}(e^{-x}-1)^2 \, dx = \int_0^a x^{s-1}(e^{-2x}-2e^{-x}+1) \, dx = \frac{a^s}{s} - 2\gamma(s,a) + 2^{-s}\gamma(s,2a), $$
in terms of the lower incompl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Pole and removable sigularity I try to solve the following problem. I do not sure how to begin :
Let $f$ be a holomorphic function on $\mathbb{C}\setminus \{0\}$. Assume that there exists a constant $C > 0$ and a real constant $M$ such that $$|f(z)| \leq C|z|^M$$ for $0 < |z| < \frac{1}{2}.$ Show that $z=0$ is either a... | Suppose first that $M\ge0$. Then $f$ is bounded on a neighborhood of $z=0$, $z=0$ is a removable singularity and $f$ has a zero at $z=0$ of order $\lceil M\rceil$.
If $M<0$ consider $g(z)=z^{\lceil -M\rceil}\,f(z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Weak convergence + compactness = strong convergence? Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for all $x^*\in X^*$ the dual space of $X$.
I know that we have the stro... | I just realized that the sequence $(x_n)_n$ has the following property:
"Every subsequence has a subsequence which converges to $x$."
It follows that the whole sequence must converge to $x$. For more details see the questions:
Question 1
Question 2
Question 3
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to draw an $405^\circ$ angle? In a math test a question was to draw a $405^\circ$ angle. Is it formally correct to say draw an angle as I think that in geometry, an angle has just some formal definition. So what is the connection between the formal definition and the drawing? And how do one draws angles over $360^\... | It depends on how you think about angles. You can either agree that $405^\circ$ is exactly the same as $45^\circ$. -- This is how mathematicians usually think about it. Or you can think about it as $1$ complete rotation ($360^\circ$) and then an additional $45^\circ$. -- This is how engineers usually think about it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Extension of Integral Domains
Let $S\subset R$ be an extension of integral domains. If the ideal $(S:R)=\{s\in S\mid sR\subseteq S\}$ is finitely generated, show that $R$ is integral over $S$.
My first attempt was to show that $R$ is finitely generated as an $S$-module, then the extension is immediately integral. Is ... | Hint. The claim follows from the standard determinant trick.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1195958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Can the empty set be an index set? I ran into a question, encountered in a computational course.
Could anyone tell me why the empty set $ \emptyset $ can be an index set?
My source is this book
| If you've seen How I Met Your Mother, you might remember the episode when Barney is riding a motorcycle inside a casino, and when the security guards grab him he points out one simple thing: "Can you show me the rule that says you cannot drive a motorcycle on the casino's floor?".
Mathematics is quite similar. If there... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proving that $\Delta(M \times M)$ is a submanifold of $M \times M$ I am struggling to prove that
$\Delta(M \times M) = \{(x,x) : x \in M\}$ is a submanifold of $M \times M$.
A manifold M is a submanifold of N if there is an inclusion map $i:M \rightarrow N$
such that:
*
*$i$ is smooth
*$Di_x$ is injective for eac... | Hint: Consider the map $M \to M \times M$ given by $x \mapsto (x,x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. How to prove that $G$ is an abelian group? Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G$ is an Abelian group. I know that the answer for this question has been already posted and I have seen it. However, could somebody explai... | Suppose that $a,b \in G$ are arbitary elements of the group G, with the assumption that $(ab)^3 = a^3 b^3$, and $(ab)^5 = a^5 b^5$.
Observe that for
$$
(ab)^3 = a^3 b^3
$$
Multiplying left and right by respective inverses yields
$\implies ababab = aaabbb \implies baba = aabb $
In addition we have
\begin{equation}
(ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196261",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why is the following collection of sets equals the following? Suppose $(A_n)$ is a sequence of events, For any $I \subset \{1,2,\ldots \} $, set
$$ C_I = \bigg( \bigcap_{n \in I} A_n \bigg) \cap \bigg( \bigcap_{n \notin I } A_n^c \bigg) $$
I am trying to show that for any $n \geq 1 $ we have
$$ \bigcup_{|I| < \infty,... | Look at how something gets into a given $C_I$. Let $I$ be a finite index set and $a\in C_I$
then we have $\forall k\in I,a\in A_k$. We also have $a\in A_j^C$ for $j\notin I$, or in other words, $a\notin A_j$ Therefore, for a given index set, we're collecting all the events which are in every one of those index set... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Smallest n to align sample mean with population mean There's a question in my book that I just do not understand. This is it in its entirety:
Let $ \bar{X} $ be the sample mean of a random sample of size $ n $ from a normal distribution with a variance of 9. Find the smallest sample size such that the sample mean is wi... | Hint:
If $X_i$~$N(\mu,\sigma)$ represent $n$ random variates, then
$$\frac1n \sum_{i=1}^{n}X_i\text{ has distribution } N(\mu, \frac{\sigma^2}{n})$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compact operators on Hilbert Space I m working on the following problem:
Let $K:H\rightarrow H$ be a compact operator on a Hilbert space. Show that if there exists a sequence $(u_n)_n\in H$ such that $K(u_n)$ is orthonormal, then $|u_n|\rightarrow \infty$.
Here is my argument: It is suffice to show for all $u\in H$, t... | I'd proceed as follows:
*
*An orthonormal set has no limit point.
*Therefore, any set containing an infinite orthonormal set is not compact.
*If $|u_n|$ is bounded, then $K$ maps a bounded set to a set whose closure is not compact.
Your proof got confused in the first step
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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inequality regarding norm of linear operator Let $T$ be a linear operator on a vector space $X$. For $x\in X$, I know there's the inequality that $$||Tx||<||T||||x||$$
Yet I'm wondering what are those norms. Are they arbitrary? especially on the right hand side there's operator norm and vector norm, how do we cooperate... | There are a few equivalent definitions of the operator norm. Perhaps the most relevant for this particular question is the following:
Let $T:V \to W$ be a linear map between two normed vector spaces $V$ and $W$. The operator norm $\|T\|_{op}$ of $T$ is defined to be
$$\|T\|_{op} := \sup \left\{\frac{\|Tx\|}{\|x\|} : x ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to determine the limit of this sum? I know that $\lim_{x\to\infty} \dfrac{2x^5\cdot2^x}{3^x} = 0$. But what I can't figure out is how to get that answer. One of the things I tried is $\lim_{x\to\infty} 2x^5 \cdot \lim_{x\to\infty}(\dfrac{2}{3})^x$, but then you'd get $\infty \cdot 0$, and I think that is undefined.... | $$F=\lim_{x\to\infty}\frac{2x^5}{(3/2)^x}$$ which is of the form $\dfrac\infty\infty$
If L'Hospital's rule is allowed,
$$F=\lim_{x\to\infty}\frac{2\cdot5x^4}{(3/2)^x\ln3/2}$$ which is again of the form $\dfrac\infty\infty$
So, we can apply L'Hospital's rule again and so on
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Why do we take $R$ when constructing the tensor algebra?l Let $R$ be a commutative ring and $M$ be an $R$-module.
Define $T^0(M)=R$ and $T^n(M)=M\otimes...\otimes M$(n-times) for $n\in\mathbb{Z}^+$.
Then we take $T(M)\triangleq \oplus T^n(M)$ and give an operation to make it an $R$-algebra.
My question is why do we tak... | You want $T(M)$ to be an $R$-algebra. Thus, the role of $T^0(M)$ is to give you an algebra homomorphism $R\stackrel{\sim}{\to} T^0(M)\to T(M)$.
When $i=0$ and $j>0$ you define $r\cdot (n_1\otimes \dots\otimes n_j)=(rn_1)\otimes \dots\otimes n_j$, similarly for $j=0$. When $i=0=j$, then the multiplication in $T(M)$ is j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Aproximation of $a_n$ where $a_{n+1}=a_n+\sqrt {a_n}$ Let $a_1=2$ and we define $a_{n+1}=a_n+\sqrt {a_n},n\geq 1$.
Is it possible to get a good aproximation of the $n$th term $a_n$?
The first terms are $2,2+\sqrt{2}$, $2+\sqrt{2}+\sqrt{2+\sqrt{2}}$ ...
Thanks in advance!
| For the third term, we have the following. Let $c_n = \sqrt{a_n} -\frac{1}{2}n + \frac{1}{4}\ln n$, then $c_{n+1} -c_{n} = -\frac{1}{2} \cdot \frac{\sqrt{a_n}}{(\sqrt{a_n}+\sqrt{\sqrt{a_n}+a_n})^2} + \frac{1}{4} \ln(1+ \frac{1}{n})\quad (1)$. Note that $a_n = \frac{1}{4}n^2 - \frac{1}{4}n\ln n + o(n \ln n)$, the first ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1196979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Every Partially ordered set has a maximal independent subset I am working on this problem:-
Every Partially ordered set has a maximal independent subset.
Definition:Let $\langle E,\prec\rangle$ be a partially ordered set. A subset $A\subset E$ is called independent set if for any two of its elements $a , b$ neither ... | This is a classical application of Zorn's lemma, but you can just as well use the Teichmüller–Tukey lemma instead.
HINT: Show that $\mathcal F=\{A\subseteq E\mid A\text{ is independent}\}$ has finite character. What can you conclude about a maximal element in $\cal F$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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linear algebra help - diagonal matrix and triangular matrix (a) Suppose that the eigenvectors of an n×n matrix A are the standard basis vectors ej for j = 1, . . . , n. What kind of matrix is A?
(b) Suppose that the matrix P whose columns are the eigenvectors of A is a triangular matrix. Does that mean that A must be ... | I suppose that the basis vectors $\mathbf{e}_i$ are proper eignevectors of $A$, and $A$ is diagonalizable. In this case you have:$A=PDP^{-1}$ where $D$ is a diagonal matrix with the eigenvalues of $A$ as diagonal elements and $P$ is a matrix that has as columns the eigenvectors, so, in your case $P=I$ and A is diagona... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's the difference between $|z|^2$ and $z^2$, where $z$ is a complex number? I know that $|z|^2=zz^*$ but what is $z^2$? Is it simply $z^2=(a+ib)^2$?
| If $z=a+ib$, then
\begin{align*}
z^2=&\,z\times z=(a+ib)\times(a+ib),\\
\left|z\right|^2=&\,z\times\overline z=(a+ib)\times(a-ib).
\end{align*}
Note that $\left|z\right|^2$ is always real and non-negative, whereas $z^2$ is, in general, complex.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Compute the limit of $\frac{\log \left(|x| + e^{|y|}\right)}{\sqrt{x^2 + y^2}}$ when $(x,y)\to (0,0)$ $$\lim_{(x,y)\to (0,0)} \frac{\log \left(|x| + e^{|y|}\right)}{\sqrt{x^2 + y^2}} = ?$$
Assuming that $\log \triangleq \ln$, then I tried the following:
1. Sandwich rule
Saying that $\log \left(|x| + e^{|y|}\right) < |x... | This is a very non-rigorous approach, but intuitive.
The Taylor expansion (about $v=1$) of $\log(v)$ is $(v-1)+O(v^2)$, and the Taylor expansion (about $u=0$) of $e^u$ is $1+u+O(u^2)$.
Thus $\log(|x|+e^{|y|})$ is approximately $|x|+e^{|y|}-1$ which is approximately $|x|+|y|+O(x^2)+O(y^2)$. In polar coordinates this is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Turn a power of $2$ into a multiple of $7$, maybe adding a $2$ but not a $7$ On St. Patrick's Day, the Lucky Charms leprechaun wagered me a bottle of Glenfiddich I couldn't solve this math problem before midnight:
There is a power of $2$ that can be turned into a multiple of $7$ with a simple rotation of a representat... | Hmm, those leprechauns are tricky...
Perhaps we need to rotate $16$ to get that multiple of 7, $91$. I can't get the rest of it to make sense, though - I tried working through other bases, no joy yet - I would have expected him to allude to going from $31_5$ to $331_5$, because they're very fond of dublin'.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 2
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Elementary proof of topological invariance of dimension using Brouwer's fixed point and invariance of domain theorems? http://people.math.sc.edu/howard/Notes/brouwer.pdf https://terrytao.wordpress.com/2011/06/13/brouwers-fixed-point-and-invariance-of-domain-theorems-and-hilberts-fifth-problem/
These papers give fairly ... | If $V$, open subset of $\Bbb R^m$, were homeomorphic to an open subset of $\Bbb R^n$, $U$, let $f: U \to V$ be a homeomorphism. (Suppose WLOG that $m \leq n$.) Compose with a linear inclusion map $\Bbb R^m \hookrightarrow \Bbb R^n$ to get a continuous injective map $U \to \Bbb R^n$ with image contained in the subspace ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $HK$ is a subgroup iff $HK=KH$. Let $H$ and $K$ be subgroups of a group $G$, and let $HK=\{hk: h
\in H, k \in K\}$, $KH=\{kh: k \in K, h \in H\}$. How can we prove that $HK$ is a subgroup iff $HK=KH$?
| It is usually helpful to recall that $ L$ is a subgroup if and only if $ L^ 2=L $, $ L^{- 1}= L $ and $ L\neq \varnothing$.
If $HK$ is a subgroup, then $HK=(HK)^{-1}=K^{ -1}H^{-1}=KH$. Hence the condition is met. Conversely,
$$ \begin{align}(HK)^{ -1} &= K^{ -1}H^{-1}\\&=KH =HK\text{, }\\(HK)(HK)=H(KH)K&=\\H( HK)K&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197732",
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Question about the relation between the adjoin and inverse of linear operator on Hilbert space I am teaching myself functional analysis from a CS background. I am clueless about the following exercise problem of introductory functional analysis. Any hint or help is appreciated. Thanks a lot!
Let $H$ be a Hilbert space ... | Hint: Since $T$ is a bounded bijective linear operator, you have the existence of $T, T^*, T^{-1}$ and $(T^{-1})^*$ (why?)
So then $\left< x, y\right> = \left<T^{-1}Tx, y\right> = \left<Tx, (T^{-1})^*y\right> = \left<x, T^*(T^{-1})^*)y\right>$, for all $x,y\in\mathcal{H}$. What does this tell us about the relationship... | {
"language": "en",
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Converting a cubic to a perturbation problem I'm trying to learn about Perturbation, but feel like I'm confused before I've even started.
Right now I'm focused on using them to find solutions to polynomial equations.
The initial example I've been given has $x^3 - 4.001x + 0.002 = 0$, the numbers clearly lend towards... | the idea, as i understand it, is to identify a small or large parameter in the problem. you have identified in yours as $\epsilon.$ the non perturbed problem, that is with $\epsilon = 0$ has easy solution. in your case this is $x^3 - 4x = 0.$ the solutions are $$x = 0 , 2, -2$$ we can three roots of the perturbed pro... | {
"language": "en",
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Why is $\ker\omega$ integrable iff $\omega\wedge d\omega=0$? Suppose $\omega$ is a nonvanishing $1$-form on a $3$-manifold $M$. It's known that $\ker\omega$ is an integral distribution iff $\omega\wedge d\omega=0$.
I'm trying to understand this, but I don't get why $\ker\omega$ integrable implies $\omega\wedge d\omega... | Yes, you can assume $X,Y\in\ker(\omega)$. Since $M$ is three dimensional $L=\bigwedge^3TM$ is a line bundle over $M$, and $\omega\wedge d\omega$ is naturally defined on (local) sections of this line bundle. Take a local basis $(X,Y)$ of vector fields parallel to $\ker(\omega)$, and a smooth local extension $Z$ of some ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1197985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Is it okay to define k-th symmetric power of $M$ in this way? I want to define the tensor algebra and related algebras in a very formal way. I will illustrate how I tried to define algebras below.
Let $R$ be a commutative ring and $M$ be an $R$-module.
Consider the tensor algebra $T(M):=\bigoplus_{n\in \mathbb{N}} M^{... | The inclusion $T^k(M) \to T(M)$ induces a map
$$
T^k(M)/I \cap T^k(M) \to T(M)/I = S(M)
$$
which is obviously injective. Your definition of $S^k(M)$ is exactly the submodule of $S(M)$ which is the image of this map (since obviously $T^k(M) \to S(M)$ is trivial on $I \cap T^k(M)$). Therefore, the two definitions give ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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stuck with root test for series convergence to $1$ for $\sum_{n=0}^{\infty} \left(\frac{(1+i)}{\sqrt{2}}\right)^n$ I've got this series and I used the series convergence root test.
However my problem is: The result of the root test is one, so I can't show wheter the series converges/diverges
$$\sum_{n=0}^{\infty} \lef... | Actually it is a geometric series with ratio $\,\mathrm e^{\tfrac{\mathrm i\pi}4}$ and the general term is periodic, hence it cannot converge.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof verification: Prove that a tree with n vertices has n-1 edges This question is not a duplicate of the other questions of this time. I want to ask is how strong is the following proof that I am going to give from an examination point of view?
Proof:
Consider a tree $T$ with $n$ vertices. Let us reconstruct the t... | What seems missing from the proof is:
*
*After the $n$-th vertex is added, how do we know there exists some "unused" vertex $v$ which is adjacent to a "used" vertex?
*How do we know that $v$ is not adjacent to multiple "used" vertices?
*How do we know that this algorithm will terminate, having used all the vertic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Error in Introduction to Mathematical Philosophy Is this an error in the text or am I reading incorrectly. What am I missing?
Introduction to Mathematical Philosophy Page 18 Definition of Number
“A relation is said to be “one-one” when, if $x$ has the relation in question to $y$, no other term $x_0$ has the same relati... | It might help to consider the examples that Russell gives in the very next paragraph:
In Christian countries, the relation of husband to wife is one-one ;
in Mahometan countries it is one-many ; in Tibet it is many-one. The
relation of father to son is one-many ; that of son to father is
many-one, but that of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $(s_n)$ be a sequence that converges... Exercise 8.9 from Elementary Analysis: The Theory of Calculus by Kenneth A. Ross:
Let $(s_n)$ be a sequence that converges.
(a) Show that if $s_n \geq a$ for all but finitely many $n$, then $\lim s_n \geq a$.
(b) Show that if $s_n \leq b$ for all but finitely many $n$, th... | I think making sense of it would involve drawing a kind of Cauchy sequence, like this typical one:
Pretty much what's going on is that, if only finitely many $s_n<a$, then when $n$ is large enough, $s_n\geq a$. This, you may recognize, is the definition of a limit (more or less). In this picture, imagine drawing a hor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $f$ such that $\int_{-\pi}^{\pi}|f(x)-\sin(2x)|^2 \, dx$ is minimal Fairly simple question that's been bothering me for a while.
Supposedly it should be simple to solve from the properties of inner product but I can't seem to solve it.
Find $f(x) \in \operatorname{span}(1,\sin (x),\cos (x))$ such that $\int_{-\pi... | The reason for writing $|f(x)-\sin(2x)|^2$ rather than $(f(x)-\sin(2x))^2$ is that one can allow the coefficients in the linear combination to be complex numbers, not just real numbers, so the square need not be non-negative unless you take the absolute value first.
Now let's see how to use the "tip" you're given:
$$
\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
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limit of a two variables recursive series I am looking at a sequence which is defined as the following:
$$a_0 = a$$
$$b_0 = b$$
$$a_{n+1}=\frac{a_n+b_n}{2}$$
$$b_{n+1}=\sqrt{a_nb_n}$$
I know that both series have $a_n\geq a_{n+1} \geq b_{n+1} \geq b_n$ for every $n \geq 1$ and therefore are monotone, bounded, and con... | Both sequences converge to the arithmetic-geometric mean of $a$ and $b$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Grobner basis and subsets Let $A$ be a subset and $I$ an ideal of polynomial ring $R=k[x_1,x_2,...,x_n]$. Is there any algorithm for deciding when $A\subseteq I$?
| Without striving for efficiency:
Use Buchberger's algorithm to produce a Groebner basis $g_1,g_2,...,g_n$ of $I$.
Let $f_1,f_2,...,f_m$ be generators of $A$. For each $f_i$ run Buchberger with $g_1,g_2,...,g_n,f_i$. If the output is again $g_1,g_2,...,g_n$ for all $i=1,2,...,m$ (as opposed to $g_1,g_2,...,g_n,h_i$ wit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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how to calculate discrete event probability? A bookstore sells children's books that belong to two publishing companies A and B and were published between years 2000 and 2004. The probabilities of a book being published by companies A and B are 0.6 and 0.4. The probability that company A published a book in years 2000,... | Do you see how you can convert all the probabilities to these values?
And does that make it easier to figure out the answers?
$$\begin{array}{|c|c|c|c|c|c|} \hline
\text{Company}& \text{2000}& \text{2001}& \text{2002}& \text{2003}& \text{2004} \\ \hline
\text{A} & .06& .12& .18& .18& .06 \\ \hline
\text{B} & .12& .08&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1198951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Testing for Uniform Convergence of the sum of an Alternating Series. I'm still trying to get used in understanding the concept behind uniform convergence, so there's another questions which I'm currently have trouble trying to answer.
Suppose there's a series $$\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{2k+1}$$ and x is... | This problem isn't quite difficult. First, the series is a power series, with radius of convergence $R=1$, to obtain this you can use Cauchy-Hadamard formula or Ratio Test.Finally you use Abel's Theorem: if $f(x)=\sum_{n=0}^\infty a_n x^n$ ($a_n,x\in\mathbb{R}$) has convergence ratio $R$, and the numerical serie $\sum_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Real equivalent of the surreal number {0.5|} I've been reading up on Surreal numbers, but have some questions.
Some equivalent real and surreal numbers.
2.5 =
{2|3} =
{{{0|}|}|{{{0|}|}|}} =
{{{{{|}}|{}}|{}}|{{{{{|}}|{}}|{}}|{}}}
0 =
{-1|1} = {-2|1} = {-2,-1|1} =
{{|0}|{0|}} = {{|{|0}},{|0}|{0|}} =
... | The number for {1/2|} is "1"...
a = {{{|}|{{|}|}}|}
numeric label for a = 1
a == "1" = True
Surreal {1/2|} represented by form {{{|}|{{|}|}}|}
is equivalent to form {{|}|} represented by name "1"
...according to python code...
from surreal import creation, Surreal
s = creation(days=7)
a = Surreal([s[1/2]],[])
name ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $\binom{2n}{n}$ is an even number, for positive integers $n$. I would appreciate if somebody could help me with the following problem
Show by a combinatorial proof that
$$\dbinom{2n}{n}$$
is an even number, where $n$ is a positive integer.
I tried to solve this problem but I can't.
| Let $S$ be all subsets of $T=\{1,2,3,\dots,2n\}$ of size $n$. There is an equivalence relation on $S$ where every equivalence class has two elements, $\{A,T\setminus A\}$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $f$ be a strictly decreasing function. Then $\int_{a}^bf^{-1}=bf^{-1}(b)-af^{-1}(a)+\int_{f^{-1}(b)}^{f^{-1}(a)}f $ I'm trying to prove the fact that if $f$ is a strictly decreasing function, then:$$\int_{a}^bf^{-1}=bf^{-1}(b)-af^{-1}(a)+\int_{f^{-1}(b)}^{f^{-1}(a)}f $$
I have already proven it for strictly increas... | The lighter region is the integral on your left-hand side.
You have $$\int_a^bf^{-1}=+U-V+W$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1199283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\sum_{1}^{\infty}\int_{n}^{n+1} e^{-\sqrt{x}} dx,$ converge or diverge? Since
$$D^{-1} e^{-\sqrt{x}} \big|_{x := u^{2}} = D^{-1} e^{-u} Du^{2} = 2D^{-1} e^{-u} u = -2(u+1)e^{-u} + C
= -2(\sqrt{x} + 1)e^{-\sqrt{x}} + C,$$
we have
$$\int_{n}^{n+1} e^{-\sqrt{x}} dx = 2(\sqrt{n} + 1)e^{-\sqrt{n}} - 2(\sqrt{n+1} + 1)e^{-\s... | $$\sum_{1}^{\infty}\int_{n}^{n+1} e^{-\sqrt{x}} dx = \int_{1}^{\infty} e^{-\sqrt{x}} dx =2\int_{1}^{\infty} e^{-t} t dt = \left[-2(t+1)e^{-t}\right]_1^\infty =\frac{4}{e} $$
Since the integral has a finite value $\frac{4}{e}$ , the sum is convergent.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Identity about a Functor I'm moving my first steps in CT and suddenly after reading about functors, this question came up in my mind:
Let $F \colon A \to B$ a functor between two categories $A$ and $B$, is it true that for any arrow $f \colon a \to a'$ in $A$,
is commutative?
According to me the definition of functor... | If $F\colon A\longrightarrow B$ is a functor from the category $A$ to the category $B$, then it associates to each object $a$ of $A$ an object $F(a)$ of $B$ and to each arrow $f\colon a\rightarrow a'$ in $A$ an arrow $F(f)\colon F(a)\rightarrow F(a')$ in $B$. This association is made in a way compatible with the identi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding $\sup$ and $\inf$ of $\frac{n^5}{2^n}$ where $n$ is natural number I'm trying to find $\sup A, \inf A$ where
$$A=\{a_n=\frac{n^5}{2^n}:n\in\Bbb{N}\}, 1\not\in\Bbb{N}$$
For $n=1$ we have $a_1 = \frac{1}{2}$, $\lim_{x\rightarrow +\infty} \frac{n^5}{2^n}=0$ and after differentiating I found out that the critical p... | Note that the critical value $\frac{5}{\ln 2}\in(7,8)$ and hence $k=\lfloor\frac{5}{\ln 2}\rfloor=7$ and $j=\lceil\frac{5}{\ln 2}\rceil=8$. But $a_7>a_8$ and so $a_1<a_2<\cdots<a_7>a_8>a_9>\cdots$ and hence $\sup\{a_n\}=a_7$ and $\inf\{a_n\}=0$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two categories sharing the same objects and morphisms Is there a natural example of two categories $\mathcal{C}$, $\mathcal{C}'$ which have the same class of objects and the same class of morphisms, including source and target maps, but different composition rules?
Of course it is easy to cook up an example, for exampl... | Consider categories whose objects are finite sets $X$ and whose morphisms $X \to Y$ are subsets of $X \times Y$. I can think of at least two interesting composition operations:
*
*Think of subsets of $X \times Y$ as $|X| \times |Y|$ matrices over the truth semiring, and perform matrix multiplication.
*Think of subs... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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there exsit postive integer $x,y$ such $p\mid(x^2+y^2+n)$ For any give the postive integer $n$,and for any give prime number $p$.
show that
there exsit postive integer $x,y$ such
$$p\mid(x^2+y^2+n)$$
My approach is the following:
Assmue that $n=1,p=2$,we choose$(x,y)=(1,2)$
$$2\mid6=1^2+2^2+1$$
Assmue that $n=1,p=3$, ... | The result is easy to prove if $p=2$, so we can assume from now on that $p$ is odd.
Modulo $p$, there are $\frac{p+1}{2}$ squares, namely the $\frac{p-1}{2}$ quadratic residues of $p$, and $0$.
So modulo $p$ there are $\frac{p+1}{2}$ distinct values of $x^2$. There are also (for fixed $n$) $\frac{p+1}{2}$ distinct valu... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $∑_{n=0}^∞c_n z^n $ be a representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$ Let $∑_{n=0}^∞c_n z^n $ be a power series representation for the function $\frac{1}{1-z-z^2 }$. Find the coefficient $c_n$ and radius of convergence of the series.
Clearly this is a power series with center $z_... | Taylor series of function is
$$1+z+2z^2+3z^3+5z^4+8z^5+13z^6+21z^7+....$$
the coefficients are Fibonacci numbers
$$F(n)=F(n-1)+F(n-2)$$
hence
$C_n=F(n)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1199807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the sum of the series $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots$ My book directly writes-
$$\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots=-\ln 2+1.$$
How do we prove this simply.. I am a high school student.
| In calculus there's this famous alternating harmonic series:
$S = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + ... + (-1)^{(n+1)} \cdot \frac{1}{n} + ... $
(*) It's convergent and its sum is equal to $ln2$
Your series is equal to exactly $T = -S + 1$ so it's
also convergent and its sum must be exactly $(-ln2+1)$
I r... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Derivative of a function defined by an integral with $e^{-t^{2}}$ I'm facing a bit of a tricky question and I can't figure out how to get to the correct answer. We have to find the derivative of the following functions:
(1) $F(x) = \int^{x}_{3} e^{-t^{2}} dt$
(2) $G(x) = x^{2} . \int^{5x}_{-4} e^{-t^{2}} dt$
(3) $H(x) ... | If $u$ and $v$ are functions in $x$ and $$F(x) = \int^{v}_{u} f(t) dt$$ , then
$$F'(x)=v'f(v)-u'f(u)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1199969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding for what $x$ values the error of $\sin x\approx x-\frac {x^3} 6$ is smaller than $10^{-5}$
Find for what $x$ values the error of $\sin x\approx x-\frac {x^3} 6$ is smaller than $10^{-5}$
I thought of two ways but got kinda stuck:
*
*Since we know that $R(x)=f(x)-P(x)$ then we could solve: $\sin x-x+\frac {... | Consider
$$
\sin(x) = x - \frac{x^3}{6} + \frac{x^5}{120} - \ldots.
$$
This series is alternating, and the terms are strictly decreasing in magnitude if $|x| < 1$. So we get
\begin{align*}
\text{estimate} - \text{ reality} &= x - \frac{x^3}{6} - \left(x - \frac{x^3}{6} + \frac{x^5}{120} - \ldots\right) \\
&=\frac{x^5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Limit with number $e$ and complex number This is my first question here. I hope that I spend here a lot of fantastic time.
How to proof that fact?
$$\lim_{n\to\infty} \left(1+\frac{z}{n}\right)^{n}=e^{z}$$
where $z \in \mathbb{C}$ and $e^z$ is defined by its power series.
I have one hint: find the limit of abs. value a... | Expand using the binomial formula: $\displaystyle \left(1+\frac{z}{n}\right)^n = \sum_{k=0}^n {n\choose k}\left( \frac{z}{n}\right)^k = \sum_{k=0}^\infty E_k^n$ where we define $\displaystyle E_k^n = {n\choose k}\left( \frac{z}{n}\right)^k$ for $k \le n$ and $= 0$ otherwise
We want $\displaystyle \sum_{k=0}^\infty E_k^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 0
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Integrating $\int_{0}^{2} (1-x)^2 dx$ I solved this integral
$$\int_{0}^{2} (1-x)^2 dx$$
by operating the squared binomial, first.
But, I found in some book, that it arrives at the same solution and I don't understand why it appears a negative simbol. This is the author solution:
$$\int_{0}^{2} (1-x)^2 dx = -\frac{1}... | There are two places where you need to be careful about the sign. The first place is when you make the derivative of the substitution (if your substitution is 1-x = t, then after making the derivative you get -dx = dt). The second place is when you change the limits of the integration due to the substitution. The lower... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
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how to show that $f(\mathbb C)$ is dense in $\mathbb C$? Let $f$ an holomorphic function not bounded. How can I show that $f(\mathbb C)$ is dense in $\mathbb C$ ? I'm sure we have to use Liouville theorem, but I don't see how.
| By contradiction, suppose that $f(\mathbb C)$ is not dense in $\mathbb C$. Therefore, there is a $z_0\in\mathbb C$ and a $r>0$ such that $f(\mathbb C)\cap B_r(z_0)=\emptyset$. Therefore $$\frac{1}{f-z_0}<\frac{1}{r}$$
and thus, by Liouville theorem $\frac{1}{f-z_0}$ is constant. This implies that $f$ is also constant w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is the following set connected given that the union and intersection is connected Suppose $U_1, U_2$ are open sets in a space $X$. Suppose $U_1 \cap U_2$ and $U_1 \cup U_2$ are connected. Can we conclude that $U_1$ must be connected??
I am trying to find a counterexample, but I failed. PErhaps it is true? Can someone h... | Suppose $A$ and $B$ form a separation of $U_1$, i.e., $A$ and $B$ are disjoint nonempty open sets such that $A\cup B = U_1$. Because $U_1 \cap U_2$ is a connected subset of $U_1$, it must be entirely contained in either $A$ or $B$ (else we would get a separation of $U_1 \cap U_2$ by intersecting $A$ and $B$ with $U_1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If two continuous maps coincide in dense set, then they are the same Suppose $f,g : R \to R $ are continuous and $D \subset \mathbb{R}$ is dense. Suppose $f(x) = g(x) $ for all $x \in D$. Does it follow that $f(x) = g(x) $ for all $x \in \mathbb{R}$??
My answer is affirmative. Suppose $h(x) = f(x) - g(x) $. By hypothes... | An other way
Let $x\in \mathbb R\backslash D$.
By density of $D$ in $\mathbb R$, there is a sequence $(x_n)\subset D$ such that $\lim_{n\to\infty }x_n=x$. By continuity of $f$ and $g$ on $\mathbb R$ and by the fact that $f(x_n)=g(x_n)$ for all $n$,
$$f(x)=\lim_{n\to\infty }f(x_n)=\lim_{n\to\infty }g(x_n)=g(x),$$
what ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solution system $3x \equiv 6\,\textrm{mod}\,\, 12$, $2x \equiv 5\,\textrm{mod}\,\, 7$ , $3x \equiv 1\,\textrm{mod}\,\, 5$ Have solution the following congruence system?
$$\begin{array}{ccl}
3x & \equiv & 6\,\textrm{mod}\,\, 12\\
2x & \equiv & 5\,\textrm{mod}\,\, 7\\
3x & \equiv & 1\,\textrm{mod}\,\, 5... | Using $\#12$ of this,
$$2x\equiv5\pmod7\equiv5+7\iff x\equiv6\pmod7\ \ \ \ (1)$$
$$3x\equiv1\pmod5\equiv1+5\iff x\equiv2\pmod5\ \ \ \ (2)$$
$$3x=12k+6\iff x=4k+2\implies x\equiv2\pmod4\ \ \ \ (3)$$
$$(2),(3)\implies x\equiv2\pmod{\text{lcm}(5,4)}\implies x\equiv2\pmod{20}\ \ \ \ (4)$$
Now safely use CRT on $(1),(4)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$H^{n}(M)$ where $M$ is compact, orientable and connected manifold I need to show that if $M$ is compact, orientable and connected manifold of dimension $n$, then $H^{n}(M) = \mathbb{R}$.
I saw that a possible proof is to take an atlas for $M$, with $U_{\alpha}$, $\alpha = 1,..N$ and $\rho_{\alpha}$ be a smooth partit... | *
*A differential $n$-form $\alpha$ is exact if there is some differential $(n-1)$-form $\beta$ such that $d\beta = \alpha$.
*Recall that $H^n(M)$ is defined by taking the closed $n$-forms and modding out by the exact $n$-forms. Fact (B) implies that integration gives you a well-defined map $H^n(M)\rightarrow \mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Series of inverse function $A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^3=s$.
I want calculate $a_5$. What ways to do it most efficiently?
| $A(s) = \sum_{k>0}a_ks^k$ and $A(s)+A(s)^{3} =s$
We know (Cauchy product): $$A(s)^{2} = \sum_{n>0}^{\infty} \left( \sum_{i=0}^{n}a_{i} a_{n-i} \right) s^n$$
And
$$A(s)^{3} = \sum_{n>0}^{\infty} \left( \sum_{j=0}^{n}a_{n-j} \left( \sum_{i=0}^{j}a_{i} a_{j-i} \right) \right) s^n$$
Hence:
$$ \sum_{n>0}^{\infty}a_ns^n+\sum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 1
} |
Group theoretic construction for permutation algorithm Consider a permutation $\sigma = [s_1, \ldots, s_n]$. The `contracting endpoints' construction for the subsequence $[s_i,\ldots, s_k]$ is given by iteratively taking the product of cycles given by the first and last elements of the sequence, successively discarding... | Assuming you compose permutations right to left, it may be helpful to observe that $(x_2,x_k,x_{k-1},\dots,x_3)(x_1,x_2,\dots,x_k)=(x_1,x_k)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is the solution of the equation $x^x=2$ rational? Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?
| Suppose $x$ is rational. Then there exist two integers $a,b$ such that $$\left(\frac{a}{b}\right)^{a/b}=2 \\ \frac{a}{b} = 2^{b/a}.$$ But that's impossible because the RHS is rational only for $a=1$, which actually makes it also integer, while with $a=1$ the LHS is non-integer for all $b>1$. Checking that $(a,b)=(1,1)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1200919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Polynomial ideals I got stuck with an exercise while preparing for my exam, and could use a hint or two to move on...
Let $f(X) = a_n X^n+a_{n-1} X^{n-1}+ \cdots +a_0 \in \mathbb{Z}[X]$ with $a_0\neq 0$
Assuming that $X\in \langle f(X)\rangle$ then prove: $n \leq 1$ and either $\langle f(X)\rangle = \langle X\rangle$ o... | $X=f(X)g(X) \Rightarrow deg f(X) + deg g(X)=1 \Rightarrow deg f(X)=0,1.$ If $deg f(X)=0,$ then $f(X)$ is a constant. Say $f(X)=\lambda \in \mathbb Z.$ In this case equating the coefficient of $X$ from both side we get that $\lambda$ is actually a unit. (This part you got right.) Now let $f(X)=aX+b, a \neq 0.$ Since $X ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
find an inverse function of complicated one Let $f:\mathbb{R}\rightarrow \mathbb{R}$: $$f(x) = \sin (\sin (x)) +2x$$
How to calculate the inverse of this function?
So far i searched a lot in the internet but i didn't find any easy algorithm to this.
What i found is just for easy function (like $f(x)=x^2$) , but no fo... | Two general methods exist, but often it is very hard to employ them with some success:
*
*integral representations as Burniston-Siewert-like representations; see: http://www4.ncsu.edu/~ces/publist.html
*series as Lagrange series: http://en.wikipedia.org/wiki/Lagrange_inversion_theorem
It is useful to remark that "s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Compute $e^A$ where $A$ is a given $2\times 2$ matrix Compute $e^A$ where $A=\begin{pmatrix} 1 &0\\ 5 & 1\end{pmatrix}$
definition
Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb R$,
$$e^{At}=\sum_{k=0}^\infty \frac{A^kt^k}{k!}\tag{1}$$
| Or, write
$A = I + N \tag{1}$
with
$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \tag{2}$
and
$N = \begin{bmatrix} 0 & 0 \\ 5 & 0 \end{bmatrix}. \tag{3}$
Note that
$IN = NI = N, \tag{4}$
that is, $N$ and $I$ commute, $[N, I] = 0$, and apply the well-known result that for commuting matrices $B$ and $C$ we have $e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluate the following indefinite integral Evaluate the integral : $$\int\frac{1-x}{(1+x)\sqrt{x+x^2+x^3}}\,dx$$
I tried through putting $x=\tan \theta$ as well as $x=\tan^2\theta$ .but I am unable to remove the square root. I also tride by putting $x+x^2+x^3=z^2$. But I could not proceed anyway...Please help...
Updat... | Hint 1: $t \mapsto (1-x)/(1+x)$
$$\begin{equation}\displaystyle\int\frac{x-1}{(x+1)\sqrt{x+x^2+x^3}}\,\mathrm{d}x = 2\arccos\left(\frac{\sqrt{x}} {x+1}\right) + \mathcal{C}\end{equation}$$
Hint 2: One can show that $t = (1-x)/(1+x)$ is it's own inverse. In other words $x = (1-t)/(1+t)$. Hence the derivative becomes.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove $c$ satisfies the integral If $f:[0,1] \to \mathbb{R}$ is continuous, show that there exists $c \in [0,1]$ such that
$$f(c)=\int_0^1 2t f(t) \text{d}t.$$
So it's pretty clear to me that I have to use Intermediate Value Theorem and Cauchy-Schwarz inequality but I can't quite get the trick done.
Any help appreciat... | Since $f$ is continuous there exist $a,b\in [0,1]$ such that $f(a)\le f(x)\le f(b), \:\forall x\in [0,1].$ Now,
$$t\in[0,1]\implies tf(a)\le t f(x)\le t f(b).$$ So
$$2f(a)\int_0^1 tdt \le 2\int_0^1 tf(t)dt\le 2f(b)\int_0^1 tdt.$$
Can you finish now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Methods to quickly compute percentages Yesterday, talking with a friend of mine, she asked me what is a quick (and – of course – correct) way to compute percentages, say $3.7 \%$ of $149$. Frankly, I was sort of dumbfounded, because I use the following two methods:
*
*either I use ratios,
*or I start to go on with... | 3.7% of 100 = 3.7
3.7% of 40 = 4*0.37 = 1.48
3.7% of 9 = 9*0.037 = 0.333
3.7+1.48+0.333=5.513
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Regarding $\lim \limits_{(x,y,z)\to (0,0,0)}\left(\frac{x^2z}{x^2+y^2+16z^2}\right)$--is WolframAlpha incorrect? $$ \lim_{x,y,z\to 0} {zx^2\over x^2+y^2+16z^2}$$
So I am trying to evaluate this limit..
To me, by using the squeeze theorem, it seems that the answer must be zero.
I trying using the spherical ... | hint: $0 \leq \dfrac{|zx^2|}{x^2+y^2+16z^2} \leq |z|$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to prove that $(z^{-1})^{-1} = z$ and $(zw)^{-1} = z^{-1}w^{-1}$? I need to prove that $(z^{-1})^{-1} = z$ and $(zw)^{-1} = z^{-1}w^{-1}$ but the only thing I can think about is to consider
$$z = a+bi, w = c+di$$
and then prove it algebraically using laws of multiplication for complex numbers. Any ideas to prove it... | Generally, $(z^a)^b=z^{ab}$, so $(z^{-1})^{-1}=z$.
We have $(zw)(zw)^{-1}=1.$ Pre-multiply by $z^{-1}$ and then $w^{-1}$ to get the result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the correct value? My confusion is:
$(-9)^{2/3} = ((-9)^{2})^{1/3} = ((-9)^{(1/3)})^{2} = 4.32$
But my calculator shows math error, and google says:
$(-9)^{2/3} = 2.16+3.74i$
| It is because it is showing one of the three possible roots, one of them being $4.32$, and the other two are $2.16 + 3.74i$ and its conjugate
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Associativity of concatenation of closed curves from $I$ to some topological spaces $X$ I'm looking for some example of closed curve such that $f*(g*h)=(f*g)*h,$ in some topological space $X$.
I tried to use $X$ like the Sierpinski space, but I can't find such closed curve.
| It depends on your definition of $f * g$ (indeed for Moore loops every triplet satisfies the equation), but let's use the most common definition: for $f, g : [0,1] \to X$,
$$(f*g)(t) := \begin{cases}
f(2t), & 0 \le t \le \frac{1}{2}; \\
g(2t-1), & \frac{1}{2} \le t \le 1.
\end{cases}$$
If you allow your space to not be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Elementary questions about polynomials and field extensions Let $$f(x)=x^2+x+1.$$ This is irreducible in $\mathbb{Z_2}[x]$, and thus $\mathbb{Z_2}[x]/(f(x))$ is a field $K$ where $(f(x))$ is a principle ideal. I don't quite understand how I find that $\overline{0}$,$\overline{1}$, $\overline{X}$, and $\overline{X+1}$ a... | In fact, $(f(x))$ is a maximal ideal, and this is equivalent to $\mathbb{Z}_2[x] / (f(x))$ being a field. Anyway, in this quotient $\bar{X}^2 + \bar{X} + \bar{1} = \bar{0}$, so, $\bar{X}^2 = \bar{X} + \bar{1}$, and by induction any power of $\bar{X}$ can be written as a $\mathbb{Z}_2$-linear combination of $\bar{1}$ an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1201993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
If $x^3 +px -q =0$ then value of $(\alpha + \beta)(\beta + \gamma)(\alpha+\gamma)(1/\alpha^2 + 1/\beta^2+1/\gamma^2)$ I am given a cubic equation $E_1 : x^3 +px -q =0$ where $p,q \in R$ so what would be value of the expression $$(\alpha + \beta)(\beta + \gamma)(\alpha+\gamma)(\frac{1}{\alpha^2} + \frac{1}{\beta^2}+\fr... | This one is easy. You need to note that
\begin{align}
\alpha + \beta + \gamma &= 0\tag{1}\\
\alpha\beta + \beta\gamma + \gamma\alpha &= p\tag{2}\\
\alpha\beta\gamma &= q\tag{3}
\end{align}
and hence the calculation of the desired expression $E$ is given by
\begin{align}
E &= (\alpha + \beta)(\beta + \gamma)(\gamma + \a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1202095",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
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